醫用流體力學
Download
1 / 87

醫用流體力學 - PowerPoint PPT Presentation


  • 145 Views
  • Uploaded on

醫用流體力學. Arterial Fluid Dynamics 邵耀華 台灣大學應用力學研究所. Physiological Fluid Dynamics. Evolution of Arterial Pressure Away from the heart. Systemic Arteries. Conduct blood flow from Left ventricle (LV) to peripheral organs Aortic valve  Aortic arch (180° turn) Geometry changes :Tapering

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' 醫用流體力學' - darren


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

醫用流體力學

Arterial Fluid Dynamics

邵耀華

台灣大學應用力學研究所

Physiological Fluid Dynamics


Evolution of

Arterial Pressure

Away from the heart


Systemic arteries
Systemic Arteries

  • Conduct blood flow from Left ventricle (LV) to peripheral organs

  • Aortic valve  Aortic arch (180° turn)

  • Geometry changes :Tapering

  • Geometry changes : Branching

  • Mechanical properties changes




Stress strain relations of rabbit s thoracic aorta
Stress-Strain relations of agerabbit’s thoracic aorta


Fluid mechanics of elastic conduct
Fluid Mechanics of Elastic Conduct age

  • Mass Conservation

  • Conservation of momentum

  • Conservation of energy


Background
Background age

  • Fundamental VariablesPressure、 Flow

  • Geometrical VariablesSize、 Thickness 、 Length、 Curvature

  • Mechanical Properties Stiffness 、Visco-Elasticity


Equations of viscous pipe flow
Equations of Viscous Pipe Flow age

  • Consider a conduct filled with incompressible fluid of density  and pressure p, let u be the only non-zero velocity component


Poiseuille s law 1840
Poiseuille age’s Law (1840)

  • Assume steady flow, u= u(r ) with no body forces, the equation of motion


Laminar poiseuillean flow
Laminar Poiseuillean flow age

  • Rate of flow through the tube

  • Mean velocity of flow

  • Shear stress at the wall


Laminar poiseuillean flow1
Laminar Poiseuillean flow age

  • Skin friction

  • Shear stress in terms of skin friction


Implication of poiseuille s law
Implication of Poiseuille age’s Law

  • Q is proportional to the fourth power of the radius.

  • Q is directly proportional to the pressure difference.

  • Q is inversely proportional to the length of the tube.

  • If the arteries becomes constricted, the bloodpressure requires to supply the blood flow adequately will risesubstantially,leading to the state of hypertension.


Optimum design of blood vessel bifurcation poiseuille s formula
Optimum design of Blood Vessel Bifurcation (Poiseuille age’s formula)

For a given pressure drop, 1% change in vessel radius results in a 4% changes in flow

Murray (1926)

Rosen (1967)

Work done

Metabolism Energy loss


Minimum cost function for optimum vessel configuration
Minimum cost function for optimum vessel configuration age

With respect to radiusa  the optimum radius

The optimum vessel radius is proportional flow to the 1/3 power, and


Optimum vessel bifurcation that with minimum cost function
Optimum vessel bifurcation that with minimum cost function age

Minimize P at the bifurcation point B

An optimum location B would be

for arbitrary movements of B.


Let B displaced along A-B direction first age

The optimum is obtained when


Again, let B displaced in the age

C-B direction

The optimum is obtained when

Similarly, displaced B along D-B direction, we find


Similarly, displaced B along D-B direction, we find age

The continuity equation gives

We find

which is often referred to as Murray’s Law

37.5°


Let a ageo denotes the radius of the aorta, and assume equal bifurcation in all generation

If the capillary blood vessel has a radius of 5 um and the radius of the aorta is 1.5 cm.

We find n=30.

The total number of blood vessel is about

230109.

Note: in fact arteries rarely bifurcation symmetrically (a1=a2).

For human, only one symmetric bifurcation.

For dog, there are none.


Pulsatile blood flow
Pulsatile Blood Flow age

  • Consider pulsatile flow in a circular vessel, p=p(x, t) and u = u(r, t)

  • For a sinusoidal flow


Pulsatile blood flow 2
Pulsatile Blood Flow(2) age

  • The general solution of the ODE in the form involves Bessel functions of complex arguments

U(r=a)=0 (non-slip)

U(r=0)=finite


Pulsatile blood flow 3
Pulsatile Blood Flow(3) age

  • Introducing Womersley number 

  • As 0, the velocity profile becomes parabolic.

  • As , viscosity is negligible U(r)=-i P/.


Analysis of blood flow using elastic theory
Analysis of Blood Flow using Elastic Theory age

  • From Poiseuille’s Law

    the flux is proportional to the pressure difference (p1-p2). However, the blood flow in veins are remarkably non-linear.

  • The flow in elastic conduct gradually attains a maximum value as the pressure difference increases and then on longer increases.


Arterial flow in elastic tube
Arterial Flow in Elastic Tube age

  • Axial velocity, v

  • Lumen area, S


Pressure diameter relationship
Pressure-Diameter relationship age

  • Let T denotes the tension of the blood vessel per unit thickness, wall thickness h, vessel radius a

  • Let ro be the radius of zero tension state, the Hooke’s Law gives elastic constant E as


Poiseuille s flow in elastic tube
Poiseuille’s flow in elastic tube age

  • Consider steady flow in elastic tube of length L, assume the tube is long and the pressure is function of axial coordinate z, let P1 and P2 denote the inlet and outlet pressure and the external pressure surrounding the tube is P0

  • Assume the flow through the tube obey Poiseuille’s law, the flow becomes


Transmission of pulse wave velocity in elastic tube
Transmission of Pulse wave (Velocity) in elastic tube age

  • Consider inviscid and incompressible fluid flow in elastic tube of lumen area A,

  • By linearizing the equations


Transmission of pulse wave velocity in elastic tube1
Transmission of Pulse wave (Velocity) in elastic tube age

  • Combining the continuity and momentum equations,

  • The wave equations

Pulse Wave Velocity (PWV)


Analysis of aortic diastolic and systolic pressure waveforms
Analysis of Aortic Diastolic and Systolic Pressure Waveforms age

  • Constitutive relationship between aortic volume and pressure where K is the volume elasticity of the aorta, and V0 is the end-systolic volume.

  • If the aorta is very soft (K is very small), let I(t) and Q(t) denote the inflow and outflow rates, we have


  • During diastole, the aortic valve is closed and there is no flow into the aorta. Hence I(t) =0. where is a non-invasive measure aortic volume elasticity. Let Td be the duration of diastolic phase, the aortic pressure (Pd) at the end of this phase or just prior to ejection is given by

  • The volume elasticity that depicts the exponential drop of aortic pressure is given by


Reynolds strouhal womersley
Reynolds #; Strouhal #; Womersley flow into the aorta. Hence I(t) =0. where

  • Reynolds number

  • Strouhal number

  • Womersley number


Flows under the action of oscillating pressure gradient
Flows under the action of Oscillating pressure gradient flow into the aorta. Hence I(t) =0. where


Wave propagation in blood vessel
Wave propagation in Blood Vessel flow into the aorta. Hence I(t) =0. where

  • Pulse wave propagation in arteries

  • A(x, t) depends on transmural pressure,

Here c is the wave propagation velocity.


For thin walled elastic tube: flow into the aorta. Hence I(t) =0. where

  • Consider the elasticity of the tube, arterial diameter  blood pressure

For a thick walled elastic tube:


Balance of force in arterial wall
Balance of Force in Arterial Wall flow into the aorta. Hence I(t) =0. where


Resonant vibration of flow in a circular tube
Resonant vibration of flow in a circular tube flow into the aorta. Hence I(t) =0. where

  • When the tube length is equal to the half wave length

  • This is called the fundamental frequency of the natural vibration.

  • Hemodynamics : Effects of Frequency on the Pressure-flow relationship of Arterial tree


Boundary conditions
Boundary conditions flow into the aorta. Hence I(t) =0. where


Pressure flow
Pressure-Flow flow into the aorta. Hence I(t) =0. where


Mean Velocity Profile flow into the aorta. Hence I(t) =0. where

(Dog Aorta)


Velocity waveform at the upper descending aorta of a dog
Velocity waveform at the upper descending aorta of a dog flow into the aorta. Hence I(t) =0. where


Effect of womersly number on the velocity distribution
Effect of Womersly number on the velocity distribution flow into the aorta. Hence I(t) =0. where


Blood pressure evolution
Blood Pressure flow into the aorta. Hence I(t) =0. where Evolution


Effect of sinusoidal pressure wave speed of various frequencies on the instantaneous aortic pressure


Distribution of Atherosclerotic Sites in Human frequencies on the instantaneous aortic pressure


Stress Concentration frequencies on the instantaneous aortic pressure

Conditions


Atherosclerotic disease at the carotid bifurcation frequencies on the instantaneous aortic pressure

Stress contours in arterial branching


What s the blood flows in reality
What frequencies on the instantaneous aortic pressure’s the blood flows in reality?

  • Unsteady

  • Non-uniform geometry

  • Bifurcations

  • Non-Newtonian

  • Viscoelastic wall

  • Fluid-solid interactions


In vitro measurement of artery pressures and flows
In vitro measurement of artery pressures and flows frequencies on the instantaneous aortic pressure


Ultrasonic flowmeters frequencies on the instantaneous aortic pressure


Electromagnetic flowmeters

Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

Electromagnetic flowmeters have existed for measurement of blood flow rate outside the body during open heart surgery.

This miniature probe is for acute and chronic, low flow measurements in small animals and rodents.

Sizes from 1 to 10 mm internal circumference


Electromagnetic flowmeters frequencies on the instantaneous aortic pressure


Electromagnetic flowmeters1
Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

  • Faraday's principle of electromagnetic induction can be applied to any electrical conductor (including blood) which moves through a magnetic field. The electromagnetic blood flowmeter is sometimes used during vascular surgery to measure the quantity of blood passing through a vessel or graft, before during or after surgery. A circular probe with a gap to fit the vessel is fitted around the vessel. This probe applies an alternating magnetic field across the vessel and detects the voltage induced by the flow via small electrodes in contact with the vessel.


Electromagnetic flowmeters2
Electromagnetic flowmeters frequencies on the instantaneous aortic pressure

  • Alternating magnetic fields (typically at 400 Hz) are used since the induced voltages are in the microvolt region and d.c. electrode potentials may cause significant errors with unchanging magnetic fields. A number of probes are required to fit the various diameters of blood vessel.

  • An alternative design carries the sensing device on the tip of a special catheter which passes inside the vessel and generates a magnetic field in the space around it and has the electrodes on its surface.


Square wave electromagnetic blood flowmeters
SQUARE-WAVE ELECTROMAGNETIC frequencies on the instantaneous aortic pressureBLOOD FLOWMETERS


Pulse oximetry
Pulse Oximetry frequencies on the instantaneous aortic pressure

  • Takuo Aoyagi(1974) developed the principle of pulse oximetry. The next year, Nihon Kohden introduced the world's first ear oximeter, OLV-5100, which used pulse oximetry to noninvasively measure saturated blood oxygen without the need to sample blood. All pulse oximeters today are based on Dr. Aoyagi's original principle of pulse oximetry.


CCA Root frequencies on the instantaneous aortic pressure


Variation of velocity waveforms across the arterial vessel

Variation of velocity waveforms across the arterial vessel frequencies on the instantaneous aortic pressure

Common Carotid Artery (CCA)


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


CCA frequencies on the instantaneous aortic pressure


Radial Artery frequencies on the instantaneous aortic pressure

0.23 cm in diameter


Brachial Artery frequencies on the instantaneous aortic pressure


Brachial Vein frequencies on the instantaneous aortic pressure


ad